An indoor running track is to be constructed with a space...

Name: Problem 85, Chapter 7: Precalculus with Limits
Uploaded: 2020-01-15T01:32:16-08:00
Duration: 3 min 25 s
Channel: Taylor Shimono
Description: An indoor running track is to be constructed with a space for exercise equipment inside the track (see figure). The track must be at least 125 meters long, and the exercise space must have an area of at least 500 square meters. (a) Find a system of inequalities describing the requirements of the facility. (b) Graph the system from part (a).

An indoor running track is to be constructed with a space for exercise equipment inside the track (see figure). The track must be at least 125 meters long, and the exercise space must have an area of at least 500 square meters. (a) Find a system of inequalities describing the requirements of the facility. (b) Graph the system from part (a).

An indoor running track is to be constructed with a space for exercise equipment inside the track (see figure). The track must be at least 125 meters long, and the exercise space must have an area of at least 500 square meters.

(a) Find a system of inequalities describing the requirements of the facility.
(b) Graph the system from part (a).

Key Concepts

Geometric Formulas for Area and Perimeter

This concept involves using established formulas to calculate dimensions like area and perimeter (or length), which are then used to build the corresponding inequality constraints. For instance, a specified minimum area or perimeter directly informs the setup of an inequality that any potential design must satisfy.

Graphical Representation of the Feasible Region

Graphing the system of inequalities helps visualize the set of all possible solutions that meet the constraints. This graphical approach is essential for understanding how the constraints interact and for identifying the solution set, which is particularly useful in optimizing design parameters or verifying that a feasible design exists.

System of Inequalities

A system of inequalities consists of multiple inequality statements that must all be satisfied by the solution. In applied problems, such as designing facilities, each inequality represents a different real-world constraint, and together they define the set of acceptable solutions, or the feasible region.

Modeling Constraints

Modeling constraints involves translating physical, spatial, or design requirements into mathematical statements using algebraic inequalities. This technique is crucial in ensuring that all specified conditions, such as minimum dimensions or areas, are accurately reflected in the mathematical formulation of the problem.

Transcript

00:01 So for this problem, we are given a diagram that looks approximately like my diagram that i sketched out in the left -hand corner.

00:08 And we're told that we want to construct an indoor running track that has a length of at least 125 meters.

00:16 So the perimeter of our track essentially must be 125 meters or more.

00:21 And the exercise space must have an area of at least 500 square meters.

00:25 So this red area in here must have an area of at least 500 square meters.

00:31 And so for part a of our problem, we want to find a system of inequalities that will describe the requirements of our facility.

00:39 So we can go ahead and start with the requirement that our track must be at least 125 meters long.

00:45 So essentially, we are just looking at, again, the perimeter of our track, so all of the black boundaries here.

00:52 And we know that this distance here on the top and bottom is going to be x.

00:57 So we can just add two of those together to get this portion of our perimeter.

01:01 And next we need to add the parameters or the circumferences of these two half circles.

01:08 So since we have two half circles, we can essentially think of this as just adding the perimeter of one full circle.

01:14 And we can see that our diameter is going to be represented by y.

01:18 So since we know that the formula for a circumference of a circle is pi times the diameter, we can just write 2x plus pi times y.

01:27 And we know this distance has to be at least 100, 125 meters...

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